Sentence examples for some vert from inspiring English sources
'⇐' Equation (4) implies that (Vert partialphi (x^ Vert _{vert cdot vert } equiv q<1) for some (vert cdot vert ).
'⇐' If (phi|_{B} in mathcal{K} B,V)) for (B=B(x_{0},delta)), then (Vert partialphi (x_{0} Vert _{vert cdot vert }<1) for some (Vert cdot Vert _{vert cdot vert }) by Corollary 1.
Now, we can assume without any loss of generality that (widetilde{P}_{a}in CJ^{mathrm {ra}}(Omegatimes B^{n},mathbb {R}^{n})); if not, then (Oinwidetilde{P}_{a} omega,x)), for some (Vert xVert= r) and every (omegainOmega), and so (P_{a}) has a fixed point or, equivalently, our problem ((Q_{varphi})) has a solution.
If for some n, (vert z_{n}vert ) lies outside the circle of radius (max {|c|,frac{2(1+|a|)}{salpha },frac{2(1+|a|)}{sbeta }}), we guarantee that the orbit escapes.
If for some n, (vert z_{n}vert ) lies outside the circle of radius (max {|c|,frac{2(1+|a|)}{salpha }}), we guarantee that the orbit escapes.
end{aligned} (3.2) (3) If for some
Moreover, if for some (eta'in K), (Vert xi Vert = Vert x- P x Vert = Vert x-eta'Vert = Vert (eta-eta')+xi Vert ), then (eta' = eta= P x)).
(3.3) Likewise, if some elements satisfy vert nabla u_{k} vert ^{p-2}ll vert nabla w_{0} vert, (3.4) then (delta u_{k}) of iterative formula (3.3) often has the property begin{aligned} biglvert nabladelta u_{k} e) bigrvert gg1.
If for some constant (d>0), (vert f^{(m)}(x) vert geq d) for (forall xin I), then (operatorname{meas}I_{h}leq ch^{frac{1}{m}}), where (c=2(2+3+cdots+m+d^{-1})). We omit the proof of Lemma 4.1.
It is natural if we consider (L pi(f(x)),y) leq L(f(x),y)) for any function f and ((x,y) in Z), which means (pi(f(x))) is more close than (f(x)) to y in some sense, as (vert yvert leq M).
